STA258 Lecture 18

12. Analysis of Variance.pdf

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  • $\begin{array}{cccc}& A& B& C\\ & 6& 10& 14\\ & 10& 8& 13\\ & 7& 12& 11\\ & 9& 10& 10\\ \text{Average}& 8& 10& 12\end{array}$
  • $12$ sales, $120\mathrm{\$}$ total. Average sale $10\mathrm{\$}$.
  • We can either predict the next sales:
    • knowing the next country
      • We can either use $SST$ as $10\mathrm{\$}$.
      • Or mean of the sales of the next country it will occur in.
      • This ignores information from other countries.
    • not knowing the next country
      • Without additional information.
      • We can predict $10\mathrm{\$}$.
      • $SST=60$
      • $\begin{array}{l}(6-10{)}^2+(10-10{)}^2+(14-10{)}^2+(10-10{)}^2\\ +(8-10{)}^2+(13-10{)}^2+(7-10{)}^2+(12-10{)}^2\\ +(11-10{)}^2+(9-10{)}^2+(10-10{)}^2+(10-10{)}^2=60\end{array}$
  • $H_0:{\mu }_1={\mu }_2={\mu }_3$
  • $H_a:$ not all ${\mu }_i$ are equal.
  • If we reject $H_0$. That means column means aren't equal. So we should use the column means to predict the next sales, based on which country it is.
  • To test we can use F Distribution. $F_{\ast }>F_{(c-1,n-c),\alpha }$ means we reject $H_0$.
  • $\begin{array}{ccccc}\text{Source}& \text{DF}& \text{Sum of Square}& \text{Mean Sum}& F\\ \text{Treatment}& c-1& SST& \frac{SST}{c-1}& \frac{MST}{MSE}\\ \text{Error}& n-c& SSE& \frac{SSE}{n-c}& \\ \text{Total}& n-1& SST+SSE& & \end{array}$
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